The moduli space of instantons on an ALE space from 3d $\mathcal{N}=4$ field theories

Abstract

The moduli space of instantons on an ALE space is studied using the moduli space of $\mathcal{N}=4$ field theories in three dimensions. For instantons in a simple gauge group $G$ on $\mathbb{C}^2/\mathbb{Z}_n$, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the affine Dynkin diagram of $G$ with flavour nodes of unitary groups attached to various nodes of the Dynkin diagram. We provide a simple prescription to determine the ranks and the positions of these flavour nodes from the order of the orbifold $n$ and from the residual subgroup of $G$ that is left unbroken by the monodromy of the gauge field at infinity. For $G$ a simply laced group of type $A$, $D$ or $E$, the Higgs branch of such a quiver describes the moduli space of instantons in projective unitary group $PU(n) \cong U(n)/U(1)$ on orbifold $\mathbb{C}^2/\hat{G}$, where $\hat{G}$ is the discrete group that is in McKay correspondence to $G$. Moreover, we present the quiver whose Coulomb branch describes the moduli space of $SO(2N)$ instantons on a smooth ALE space of type $A_{2n-1}$ and whose Higgs branch describes the moduli space of $PU(2n)$ instantons on a smooth ALE space of type $D_{N}$.